Table 1.

Regulation factors for several different regulatory motifs.

Case	Regulation factor (Freg)
1. Simple repressor
	(1 + r)−1	${\left(1+\frac{[R]}{K_R}\right)}^{-1}$
2. Simple activator
	$\frac{1+a{e}^{-\frac{{\epsilon }_{ap}}{k_{B}T}}}{1+a}$	$\frac{1+\frac{[A]}{K_A}f}{1+\frac{[A]}{K_A}}$
3. Activator recruited by a helper (H)
	$\frac{1+a\frac{1+{he}^{-\frac{{\epsilon }_{ha}}{k_{B}T}}}{1+h}{e}^{-\frac{{\epsilon }_{ap}}{k_{B}T}}}{1+a\frac{1+{he}^{-\frac{{\epsilon }_{ha}}{k_{B}T}}}{1+h}}$	$\frac{1+\frac{[H]}{K_H}+\frac{[A]}{K_A}f+\frac{[A]}{K_A}\frac{[H]}{K_H}f\omega }{1+\frac{[H]}{K_H}+\frac{[A]}{K_A}+\frac{[A]}{K_A}\frac{[H]}{K_H}\omega }$
4. Repressor recruited by a helper (H)
	${\left(1+\frac{1+{he}^{-\frac{{\epsilon }_{hr}}{k_{B}T}}}{1+h}r\right)}^{-1}$	$\frac{1+\frac{[H]}{K_H}}{1+\frac{[H]}{K_H}+\frac{[R]}{K_R}+\frac{[R]}{K_R}\frac{[H]}{K_H}\omega }$
5. Dual repressors
	(1 + r1)−1 (1 + r2)−1	${\left(1+\frac{[R_1]}{K_{R_1}}\right)}^{-1}{\left(1+\frac{[R_2]}{K_{R_2}}\right)}^{-1}$
6. Dual repressors interacting
	${\left(1+r_1+r_2+r_1{r}_2{e}^{-\frac{{\epsilon }_{r_1{r}_2}}{k_{B}T}}\right)}^{-1}$	${\left(1+\frac{[R_1]}{K_{R_1}}+\frac{[R_2]}{K_{R_2}}+\frac{[R_1]}{K_{R_1}}\frac{[R_2]}{K_{R_2}}\omega \right)}^{-1}$
7. Dual activators interacting
	$\frac{1+a_1{e}^{-\frac{{\epsilon }_{a_{1}p}}{k_{B}T}}+a_2{e}^{-\frac{{\epsilon }_{a_{2}p}}{k_{B}T}}+a_1{a}_2{e}^{-\frac{{\epsilon }_{a_{1}p}+{\epsilon }_{a_{2}p}+{\epsilon }_{a_1{a}_2}}{k_{B}T}}}{1+a_1+a_2+a_1{a}_2{e}^{-\frac{{\epsilon }_{a_{1}p}+{\epsilon }_{a_{2}p}}{k_{B}T}}}$	$\frac{1+\frac{[A_1]}{K_{A_1}}{f}_1+\frac{[A_2]}{K_{A_2}}{f}_2+\frac{[A_1]}{K_{A_1}}\frac{[A_2]}{K_{A_2}}{f}_1{f}_2\omega }{1+\frac{[A_1]}{K_{A_1}}+\frac{[A_2]}{K_{A_2}}+\frac{[A_1]}{K_{A_1}}\frac{[A_2]}{K_{A_2}}\omega }$
8. Dual activators cooperating via looping
	$\frac{1+a_1{e}^{-\frac{{\epsilon }_{a_{1}p}}{k_{B}T}}+a_2{e}^{-\frac{{\epsilon }_{a_{2}p}}{k_{B}T}}+a_1{a}_2{e}^{-\frac{{\epsilon }_{a_{1}p}+{\epsilon }_{a_{2}p}+F_{\mathit{loop}}}{k_{B}T}}}{(1+a_1)(1+a_2)}$	$\frac{1+\frac{[A_1]}{K_{A_1}}{f}_1+\frac{[A_2]}{K_{A_2}}{f}_2+\frac{[A_1]}{K_{A_1}}+\frac{[A_2]}{K_{A_2}}{f}_1{f}_2\omega }{\left(1+\frac{[A_2]}{K_{A_2}}\right)\left(1+\frac{[A_1]}{K_{A_1}}\right)}$
9. Repressor with two DNA binding units and DNA looping
	${\left(1+r_m+\frac{r_m}{1+r_a}{e}^{-\frac{\mathrm{\Delta }{\epsilon }_{r_{a}d}+F_{\mathit{loop}}}{k_{B}T}}\right)}^{-1}$	$\frac{1+\frac{[R]}{K_a}}{\left(1+\frac{[R]}{K_m}\right)\left(1+\frac{[R]}{K_a}\right)+\frac{[R][L]}{K_m{K}_a}}$
10. N non-overlapping activators and/or repressors acting independently on RNAP
	Freg1 · Freg2 ·•••·FregN	Freg1 · Freg2 ·•••·FregN

Regulation factors for several different regulatory motifs. In the schematics of the motifs appearing in the first column, the inverted ‘T’ symbol indicates repression, arrows represent activation, and a dashed line is for DNA looping. The second column gives the regulation factor in terms of the number of transcription factors (TFs) in the cell and their binding energies, and the third column provides a translation of the regulation factor into the language of concentrations and equilibrium dissociation constants (used in the following paper [1^••]). For an arbitrary TF we introduce the following notation: in the second column, x is the combination $\frac{X}{N_{NS}}{e}^{-\mathrm{\Delta }{\epsilon }_{xd}/k_{B}T}$, and [X] in the third column denotes the concentration of transcription factor X. K_X = [X]/x is the effective equilibrium dissociation constant of the TF and its operator sequence on the DNA. Furthermore, in the third column we introduce f = e^{−ε_xp/k_BT} for the ‘glue-like’ interaction of a TF and RNAP, and ω = e^{−ε_x1x2/k_BT} for the interaction between two TFs. In cases 8 and 9, F_loop is the free energy of DNA looping, ω in case 8 is defined as e^{−F_loop/k_BT}, while [L] in case 9 is the combination $\frac{N_{NS}}{V_{\mathit{cell}}}{e}^{-F_{\mathit{loop}}/k_{B}T}$, V_cell being the volume of the cell.